Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4×C5 |
| direct product | C23×C4×#1044 |
| Fitting | C23×C4×C52 ⋊ C4 |
| derived, Sylow | C52 ⋊ C23×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | #1044 |
| commutator, hypocenter | C52 | C23×C42 |
| Frattini | C2 | C24×#1044 |
| Fitting | C23×C4×C52 | C4 |
| socle | C24×C52 | C2×C4 |
| prime radical | C24×C5⋊D5 | C22 |
| cosocle | C2×C5⋊D5 | C25 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23×C4 | |
7362 conjugacy classes of subgroups; only the 1854 normal subgroups are shown.
This group has 7362 conjugacy classes of subgroups — too many to draw. Only its 1854 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(11,12,15,13,14)(16,17,18,20,19);(3,4);(12,13,14,15)(17,18,19,20);(7,8,9,10);(1,2)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C4×#87830 |
| maximal normal subgroup of | C22×C4×#87830 |
| maximal quotient of | C22×#13104814, #72242691 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(11,12,15,13,14)(16,17,18,20,19), (3,4), (12,13,14,15)(17,18,19,20), (7,8,9,10), (1,2));